No, it isn't natural. Native English speakers do not say this to their guests. In fact, it might sound quite rude. It might be seen as either an insult to the guest or to the food : for example, you might be suggested that your guest is greedy or that you want to get rid of as much food as possible. 'Help yourself!' or - more informally - 'Tuck in!' are natural phrases that we use when we encourage guests to start eating. After the guests have eaten one helping of food, it's normal to invite them to have some more.

How about ''Eat as much as you want!' (There's plenty)'?

No, 'eat a lot' is not natural. I think we just say 'help yourself' and maybe 'have some more' later maybe.. but I think people usually eat a lot so we don't have to say it! :)

Yes, it is completely natural to say this. Another way would be “You should eat a lot” or “Please eat a lot”. I hope this helps:)